12.1.13: Diffraction by Planes of Atoms

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12.17.jpg — Figure 12.7: Diffraction from 2D (rows of atoms) to 3D (planes of atoms)

A regularly spaced line of atoms provides a simple model for describing diffraction (Figure 12.17a). However, in two dimensions crystals contain multiple identical unit cells. So, crystals have many identical lines of atoms, all causing diffraction at the same angles (Figure 12.17b). In three dimensions, a crystal contains many identical 3D unit cells, all containing planes of atoms that cause diffraction at the same angles (Figure 12.17c). A distance d_hkl separates the planes of atoms, and they diffract at angles we can calculate with Bragg’s Law. Although not clearly shown in Figure 12.17c, diffracted X-rays are not constrained to a plane but may go in any direction in 3D space.

12.18.png — Figure 12.18: Miller indices and the distance between planes

As discussed in Chapter 11, crystallographers use Miller indices (hkl) to describe planes of atoms in crystals. d_hkl is the perpendicular distance between (hkl) planes. Let’s consider planes perpendicular to this page.

Figure 12.18a shows a single (120) plane. Every unit cell is identical, so Figure 12.18b shows an entire family of (120) planes separated by the perpendicular distance d₁₂₀. Different families of planes, with the same orientation but different indices, have different d-values. Figure 12.18c shows the family of (240) planes; they parallel (120) but are half as far apart. For all crystal systems, d₁₂₀ is twice d₂₄₀. Planes with higher numbers in their Miller indices are closer together than planes with lower numbers, and in general:

d_hkl = n d_h´k´l´

where h´ = nh, k´ = nk, and l´ = nl.

The relationship between Miller indices, unit cell lengths (a, b, c), and d-values depends on the crystal system. Box 12.1 gives equations relating unit cell parameters to d-values. Derivation of equations for orthogonal systems (cubic, tetragonal, and orthorhombic) is simple compared with derivations for the other systems. For orthogonal systems:

d₁₀₀ = a
d₀₁₀ = b
d₀₀₁ = c

12.19.png — Figure 12.19: d-values and unit cell parameters

For the other systems, we must also consider the angles α, β, and γ. Figure 12.19 shows this by comparing d-values with unit cell lengths for square, rectangular, and monoclinic cells. For the orthogonal cells, d₁₀₀ = a, but for the monoclinic cell it does not.

Calculating d-Values from Cell Parameters and Miller Indices